We update our previous search for trapped magnetic monopoles in LHC Run 2 using nearly six times more integrated luminosity and including additional models for the interpretation of the data. The MoEDAL forward trapping detector, comprising 222 kg of aluminium samples, was exposed to 2.11 fb −1 of 13 TeV proton–proton collisions near the LHCb interaction point and analysed by searching for induced persistent currents after passage through a superconducting magnetometer. Magnetic charges equal to the Dirac charge or above are excluded in all samples. The results are interpreted in Drell–Yan production models for monopoles with spins 0, 1/2 and 1: in addition to standard point-like couplings, we also consider couplings with momentum-dependent form factors. The search provides the best current laboratory constraints for monopoles with magnetic charges ranging from two to five times the Dirac charge.